Substitutability in R&D and Open Innovation: Why Competing Firms Face a Paradox of Openness?

Abstract

We explore how substitutability between internal and external R&D influences business conduct with regard to R&D approaches and their implementation with the support of managerial resources in a competitive innovation game. We develop a duopoly model which incorporates the assumption of substitutability due to managerial diseconomies of scope in firms using different innovation mechanisms to access technology. We establish that firms substitute internal sources of innovation for external sources as exogenous spillovers increase to manage the diseconomies of scope. We find a tension between the knowledge disclosures and the protections of innovation returns that arises as a paradox for firms. The finding of a positive association between the choice of a firm to be connected with the R&D environment and the appropriability of its innovation returns is understood to be consistent with empirical results of a paradox of openness that firms open to external sources of innovation face.

Introduction

Substitutability is one case of how internal and external R&D strategies interact in the generation of innovation. Substitutability between two innovation activities means that the marginal return to one activity decreases as the intensity of the other increases (Cassiman & Veugelers, 2006; Milgrom & Roberts, 1990, 1995; Topkis, 1998). Two sources of substitutability between internal and external R&D are switching costs (Rothaermel & Hess, 2007) and diseconomies of scope (Hess & Rothaermel, 2011). Substitutability in innovation can be caused by switching costs between different innovation modes (internal R&D, external knowledge sources) and diseconomies of scope associated with using different innovation mechanisms to access technology in tandem. The transaction cost literature tends to view the alternative scenarios of internal knowledge generation and external knowledge sourcing for innovation as substitutes (Williamson, 1985; Pisano, 1990). This is because of its emphasis on the relative costs of conducting innovation activities in-house or externally. Such relative costs determine whether it is efficient for firms to develop innovation internally or search for knowledge externally.

“Open innovation” (OI), a term introduced in 2003 by Chesbrough (2003), involves the “use of purposively inflows and outflows of knowledge to accelerate internal innovation and expand the market for external use of innovation, respectively” (Chesbrough, 2006). A firm’s choice to be open to the external environment is related to its appropriability strategy, i.e., its approach to capturing rents due to R&D. A major problem associated with assessing and using external sources of knowledge has to do with the fact that, in order to obtain knowledge, the firm has to disclose some parts of its own knowledge to external actors (Laursen & Salter, 2014). This introduces what the authors describe as the paradox of openness, where managers make their firm open by relying on external sources in order to innovate, but also have to protect their own firm’s knowledge from being copied and appropriate the benefits derived from their firm’s innovation.

Several articles extended recent work on this paradox in OI (e.g., Alexy et al., 2013; Laursen & Salter, 2014; Ritala et al., 2015; Arora et al., 2016) in various directions. Adopting an external partner’s perspective, Wang et al.’s study (2017) reconsiders the paradox of openness by analyzing how a focal firm’s product experience and patenting experience affect an external partner’s tendency to provide external R&D services to the focal firm. Their study found support for the predictions that a firm’s product experience increases the focal firm’s external R&D sourcing, but that a firm’s patenting experience decreases the focal firm’s external R&D sourcing. Lauritzen and Karafyllia (2019) propose to shift the conceptual frame from looking at the tensions between control and openness as problems to looking at them as synergies. Their article explains how OI paradoxes unfold in and across the key themes, i.e., attracting, incorporating, and commercializing, and can be managed in practice. Research prior to Foege et al. (2019) has examined the paradox of openness from the perspective of the seeking firm, focusing on the firm-level challenges of inbound OI. In their article, Foege et al. complement that research by illuminating the tensions between sharing and protecting in individual-level outbound OI, where they consider that the paradox of openness is most prevalent, yet least understood.

The existing innovation literature proposes different major measures that firms can employ to solve the dilemma of innovative search and knowledge protection. Clarifying the relationship between knowledge and ideas, the paper by Ruiz-Pava and Forero-Pineda (2020) finds that search strategies of firms are more effective for innovation depending on the target market. Firms reporting products new to international markets exhibit search strategies combining ideas from internal sources with ideas from other firms, whereas firms reporting products new to local market reveal a search strategy centered on ideas from other firms. Tensions between generating and appropriating value may emerge in OI collaborations, but according to Stefan et al. (2022), such tensions have been scarcely explored at the micro-level. Their study bridges this gap by examining individuals’ affective responses to tensions and their outcomes in OI, thereby capturing the micro-foundations of the paradox of openness.

The main objective of this paper is to explore how substitutability between internal and external R&D influences the business conduct of innovative firms with regard to R&D approaches and their implementation with the support of managerial resources in a competitive innovation game. We seek to shed some light on the unresolved question if and how appropriability and OI are associated with each other. The empirical justification for the notion of sourcing knowledge internally and externally as substitute mechanisms to advance innovation is provided by the evidence reviewed below. To this purpose, we develop a duopoly model which incorporates the assumption of substitutability arising between internal and external R&D due to managerial diseconomies of scope in firms using different innovation mechanisms to access technology, and with that, we extend the previous analysis on strategic R&D investments with exogenous spillovers and endogenous absorptive capacity. To our knowledge, this study is among the first to investigate this topic in a game-theoretic setting. Due to the assumption that sourcing knowledge internally and externally is a substitute mechanism, the results of our model are qualitatively different from those of earlier models of R&D with exogenous spillovers or endogenous absorptive capacity determined by R&D approaches. Empirical evidence exists that provides support for our theory.

The main conclusions we reach are the following. We establish that firms substitute internal sources of innovation for external sources as exogenous spillovers increase to manage the diseconomies of scope associated with sourcing for knowledge both internally and externally. We focus on the generality of R&D performed by a firm as a way to take knowledge from external sources. We find a tension between the knowledge disclosures resulting from the R&D generality of a firm’s research approach and the protections of innovation returns facilitated by the R&D specificity of its research approach that arises as a paradox for firms. The finding of a positive association between the choice of a firm to be connected with the R&D environment and the appropriability of its innovation returns is understood to be consistent with empirical results of a paradox of openness that firms open to external sources of innovation face.

The key results in our theoretical model appear to be consistent with the evidence gathered and analyzed in a large amount of empirical studies of R&D investments and spillovers posing different research questions. We found no previous game-theoretic study usually making no built-in assumption on the relationship between the use of external sources of knowledge and internal R&D that were validated by that kind of evidence (i.e., it is usually assumed that internal and external technology sourcing activities are independent from each other) . Overall, our theoretical framework provides an inclusive interpretation of fragmentary and sometimes contradictory evidence suggesting the importance of the hypothesis of substitutability between internal and external R&D in the analysis of business conduct of innovative firms.

The remaining of the paper is organized as follows. In the next section, brief reviews of literature on substitutability in innovation, and the paradox of openness are conducted. In the “Method” section, the game-theoretical model of R&D is briefly presented and analyzed. In the “Results” section, the comparative-statics implications of the competitive innovation game are established. In the “Discussion” section, the results concerning firms’ R&D approach behavior under R&D competition are discussed. Finally, the “Conclusion” section concludes the paper with public policy implications and limitations of the study and directions for future research. A detailed description of the model, its formal analysis, and proof are contained in the Appendix.

Literature Review

A considerable number of empirical studies found hard evidence of a substitution effect between internal R&D and external technology sourcing activities. The two innovation strategies for sourcing external knowledge are technology acquisition on the market and cooperation in R&D. Serrano-Bedia et al. (2018) provide a review of the empirical studies on the nature of the relationship between different innovation knowledge sources, i.e., complementary or substitute relationships between available knowledge sources for a firm (internal, contractual arrangements in the market, and cooperation). The authors find numerous studies providing rigorous evidence on the substitutive relationship between internal and external knowledge sources. Given the accumulation of empirical evidence in favor of the substitutability hypothesis between internal and external knowledge sources, it is surprising that little attention has been devoted to its theoretical and public policy implications.

Laursen and Salter (2006) presented some empirical evidence in favor of the hypothesis that searching for knowledge that firms access internally and externally is a substitute activity, and subsequent work has produced considerably more. The importance of substitutability in R&D strategy is confirmed by Tsai and Wang (2009) and Berchicii (2013). The common features of these three papers are that a firm’s innovation performance is derived from using internal and external sources with firms tapping into external sources of knowledge through R&D contracting, and the substitutability between internal and external sources is measured through product innovation. Other papers studying the existence of substitutability on innovation performance between some innovation knowledge sources (internal, external (contracted R&D), and/or cooperation) employing different measures of innovation performance present evidence that suggest that this is indeed the case. The results from the studies by Beneito (2006) and Hagedoorn and Wang (2012) confirm previous evidence for substitutability between internal and external sources of knowledge for incremental product innovations (utility models) and for product and process innovations (patents), respectively. Love and Roper (1999, 2001), Jirjahn and Kraft (2011), and Guisado González et al. (2014) find evidence of substitutability in product innovation performance when firms combine in their R&D strategies internal and cooperation innovation knowledge sources.

The empirical literature on the nature of interaction between internal and external knowledge sources in innovation development is inconclusive, producing mixed results (Ennen & Richter, 2010; Hagedoorn & Wang, 2012). Thus, while some studies provide support for the complementarity hypothesis between internal development and external knowledge sourcing, other studies suggest the existence of a substitutability relationship between them. Still, other studies found no conclusive empirical evidence on internal and external R&D activities and their effects on innovation performance for product innovations (Schmiedeberg, 2008) and for process innovations (Vega-Jurado et al., 2009; Krzeminska & Eckert, 2016). The idea that firms benefit from complementing internal with external knowledge sources is well accepted in the literature (e.g., Arora & Gambardella, 1990; Cassiman & Veugelers, 2006), but most studies on complementarity of R&D activities focus on product innovations rather than process innovations (Krzeminska & Eckert, 2016). New empirical evidence suggests the likely existence of a contingent relationship between internal and external knowledge sources in shaping a firm’s innovative output instead of a clear-cut answer to the question of whether different innovation knowledge sources are complementary or substitute factors (Serrano-Bedia et al., 2018).

Several organizational literature studies argued that organizational attention is a valuable and scarce resource in organizations (Simon, 1957; Cyert & March, 1963; March & Simon, 1993). Organizational scholars noticed the importance of attention as a scarce resource for determining firm behavior. Managers deal with one or a few problems or aspects of reality at a time, because the limits on attention simply do not allow everything to be taken into account at once (Simon, 1947). The limited span of attention of any individual manager limits the ability of a firm to pursue many external linkages for innovation (Love et al., 2013). Firm behavior, including decision-making and strategy formation, is the result of how firms channel and distribute the attention of their decision-makers (Ocasio, 1997, 2011). Attentional perspective or focus on a limited number of issues and answers is a critical determinant of strategic action and adaptation. Firm innovation from a managerial perspective can be regarded as a constrained optimization problem (Rothaermel & Hess, 2007). In high-technology industries, firms face essentially limited managerial resources. This problem is particularly important when different innovation mechanisms can be substitutes for one another. Their simultaneous use by a firm might imply a decrease of its innovative output at the margin. It is sufficient for a substitutive relationship to take place that using one innovation activity marginally decreases the benefit from using another (Hess & Rothaermel, 2011). The reason for this argument stems from the potential diseconomies of scope associated with using different mechanisms to access knowledge and innovate. Diseconomies of scope can be motivated by managerial constraints (Belleflamme & Peitz, 2015).

Firms cannot realize the benefits from external knowledge merely by being exposed to it (Cohen & Levinthal, 1989, 1990). Instead, they must develop “absorptive capacity.” R&D is important to build absorptive capacity since the ability to identify, assimilate, and exploit new external knowledge is a function of own R&D. Intensity of R&D effort is crucial to put an effective absorptive capacity in place. The role of external linkages in building absorptive capacity has been increasingly recognized (Lim, 2009). Establishing absorptive capacity through connectedness to external sources of knowledge can help a firm to access and acquire knowledge from those sources. A firm’s effort to build absorptive capacity involves not only R&D expenditures but also how it manages its internal R&D together with efforts to link internal and external R&D. Organizational attention allocated to searching any particular source of existing ideas and information plays an enabling role by allowing the organization to build its absorptive capacity for innovation (Koput, 1997). If the attention allocated to an inside or outside source of ideas is very low, the organization may not be able to develop absorptive capacity by not having accumulated sufficient information. Absorptive capacity can be framed as an attention directing and action-generating capability which applies to knowledge residing within or across firms (Kim et al., 2016). Innovation becomes possible when the firm allocates attention to the recognition, assimilation, and exploitation of new knowledge held externally or internally to the firm.

Firms open to external knowledge sources consistently face the paradox of sharing and protecting knowledge at the same time (Bogers, 2011; Alexy et al., 2013; Ritala et al., 2015). Laursen and Salter (2014) call this the paradox of openness. Arora et al. (2016) argue that both patenting (appropriability) and external sourcing (openness) are jointly determined strategic choices made by firms. Until then, the existing empirical literature has treated the trade-off between appropriability and openness as a causal relationship. Openness, or external sourcing, and the paradox that it creates entails a trade-off when opening up to outside sources of knowledge to create innovations may weaken the firm’s power to capture rents from that knowledge. Associated with this paradox, there are two conflicting views in the literature, which are called the “spillover prevention” theory and the “organizational openness” theory. Arora et al. (2016) expect to see a positive association between patenting and openness in the first view and an inverse relationship in the second one. The relationship between appropriability and OI is an unresolved question (West, 2006). On the one hand, stronger intellectual property regimes are directly associated with more OI. A high degree of patent protection makes patents more attractive for technology licensing purposes and promotes vertical specialization (Gallini, 2002). On the other hand, the case of open-source software raises questions about this relationship. Open-source software is an increasingly popular example of OI. The free and open-source movements contain intellectual property restrictions intended to force sharing of software and value free access to source code (West & Dedrick, 2005). Firms involved in open-source software often make investments that will be shared with real and potential rivals (West & Gallagher, 2006).

In order to explore this paradox of openness, several empirical investigations were conducted. Some studies find a positive relationship between appropriability and openness. Based on empirical analysis of data from 2005 UK Innovation Survey, Laursen and Salter (2014) explore how firms’ degree of openness for innovation is related to the strength of their appropriability strategy. The measure of the overall strength of the firm’s appropriability strategy combines responses on the degree of importance to the firm of different methods of protection. It includes formal methods, such as patents, registration of design, and trademark, as well as informal methods, such as secrecy, lead time, and product complexity. The aspects of firm openness examined are the breadth of the firm’s search efforts and the range of types of partner organizations in formal collaborations for innovation. The authors find that overall appropriability strategy has a concave relationship with external search breadth and innovation collaboration breadth. That is, there is a concavity association (at least in terms of decreasing positive marginal effects) between appropriability and openness in the entire appropriability strategy scale. Their models confirm the predominantly positive relationship between strength of the appropriability strategy on the one hand and breadth of external search and collaboration on the other.

Using data drawn from the 2003 Innovation in Australian Business Survey, Huang et al. (2014) investigate how openness actually affects a firm’s management of appropriability regimes to recoup benefits from its innovation. The authors find that the degree of openness is curvilinearly related to the scope of appropriability regimes. Firms with a higher degree of openness tend to employ a larger range of appropriability regimes. While firms that are more open may use an increasing range of appropriability regimes, there comes a point where the degree of openness has a declining marginal effect on the dependent variable. The degree of openness is measured by the breadth of external knowledge sources and the scope of inter-organizational collaborations that a firm is involved with, and the measure of the scope of appropriability regimes combines both formal and informal methods of intellectual property protection. Formal methods include patent and copyright or trademark, and informal methods include secrecy and complexity of product design.

Arora et al. (2016) use data from the 2012 Survey of Innovation and Patent Use and the 2009 UK Community Innovation Survey to assess the link between patenting and openness (external sourcing). The openness is defined by the number of different types of the firms’ external collaborators. Firms cooperate in innovative activities with organizations such as suppliers of equipment, clients or customers, competitors, and universities and other higher education institutions. The results show a positive association between patenting and openness. The relationship between patenting and collaboration in innovation is stronger for technology leaders than for followers. The authors treat both patenting and external sourcing (openness) as jointly determined decisions made by firms. The increase in patenting due to openness is higher for leading firms than for followers. Zobel et al. (2016) analyze how the patent stock of new entrants in industries shaped by systemic innovation influences their subsequent openness in innovation. Openness in innovation is measured by the number of inter-organizational relationships that a firm enters in a given year. The results suggest that patents have a strongly positive effect on technology-intensive relationships that focus on joint scientific research, while relationships that are of decreasing technology intensity are gradually less affected by the new entrant’s patent stock.

Model

In this section, we present a description of the model which is centered on the formulation of the effective R&D level. We consider the fully non-cooperative version of a game of R&D with exogenous spillovers and endogenous absorptive capacity where all strategic variable research approach, managerial resource allocation, R&D output, and output-quantity are chosen independently. The detailed representation of the remaining part of the model as well as the derivation of its equilibrium results is provided in the Appendix.

A simple duopoly model can be set up to highlight the role of substitutability between the innovation mechanisms of sourcing new knowledge internally and externally. The market consists of two firms 1 and 2, indexed by i and j, where i ≠ j. Firms participate in a three-stage game of R&D competition. Firms act independently at each stage of the game. Firms first simultaneously choose research orientations and allocate scarce managerial resources to search for new knowledge internally and externally. Each of the two firms has to make two decisions in stage 1. The remaining of the timeline of the duopoly game is that firms choose second their R&D output levels and finally their Cournot outputs. The intuition behind this timeline is that the choice of a R&D approach and the allocation of managerial resources to different mechanisms of technology sourcing involve a longer-term commitment than choosing an R&D output level.

Managerial attention plays an important role in identifying important ideas for innovation and absorption. The quantity of relevant ideas to be implemented in the production site of a firm is a function of the amount of attention allocated to search activities and the number of new ideas discovered by all firms. The amount of attention allocated to internal and external sources of knowledge is required to implement the firm’s chosen R&D orientation. Firms use different R&D strategies, choosing between innovate and imitate. Firms can improve their market profitability by making R&D investments that reduce the unit costs of production. We posit that the effective level of unit-cost reduction of firm i is given as

$${X}_i=\mathit{\operatorname{Min}}\left\{1,{r}_{ii}\right\}{x}_i+\mathit{\operatorname{Min}}\left\{{\delta}_i{\delta}_j,{r}_{ij}\right\}\beta {x}_j,i=1,2,i\ne j.$$

(1)

The effective cost reduction of firm i depends not only on its own knowledge produced, x_i, but also on the other firm’s knowledge generated, x_j, via intermediate or final spillovers respectively, i, j = 1, 2 and i ≠ j. R&D activities are associated with positive spillovers. A firm’s R&D output leaks out to its rival at an exogenous rate 0 ≤ β ≤ 1, but shared R&D output is only partly assimilated and absorbed by the rival firm. Decision-makers of firm i allocate the amount r_ii of its scarce managerial resource to look for innovative ideas within the organization and the amount r_ij to look to see what others are doing. In the firm’s effective R&D activity, r_iix_i refers to the cost reduction obtained by innovation efforts r_ii provided that 1 > r_ii. The marginal productivity of firm ’s managerial effort to imitate r_ij is assumed to be constant too, either positive or zero. Beath et al. (1998) introduce a two-stage process whose first R&D output is knowledge produced and final R&D is lower unit costs. Corresponding to the first stage is a process of search and discovery, and corresponding to the second stage is a process of combining pieces of knowledge into a specific form. Knowledge serves as an input that determines the amount of R&D output at the second stage.

The benefits arising from the utilization of external knowledge reflect the absorption efforts that make possible the imitation of the rival’s ideas and are conditional upon the connectedness with the R&D environment pursued by firms. The absorptive capacity of a firm is dependent upon the firms’ choices of R&D approaches 0 ≤ δ_i ≤ 1 and 0 ≤ δ_j ≤ 1 by firms i and j, respectively. Building this form of absorptive capacity involves a significant amount of managerial attention. The degree of firms’ connectedness pursued makes cognitive resources and absorption costs critical to the imitation of a fraction of external knowledge. We define the degree of connectedness between firms as the product of firms’ R&D designs, δ_iδ_j, which is consistent with the way firms’ connectedness is characterized by Wiethaus (2005). Higher values for r_ij are likely to bring about an increase in the effective degree of connectedness Min{δ_iδ_j, r_ij}. Clearly, the consistency requirements 0 ≤  Min {∙, r_ii}, Min {∙, r_ij} ≤ 1 imposed on the extent to which knowledge generated is transformed through managerial efforts into unit cost reduction are satisfied.

Firms face a finite capacity to allocate attention of decision-makers to search activities. We suppose that the supply of managerial resources to a firm is fixed at a given amount. We assume that the tightly constrained resource endowment of firm i is equal to 1:

$${r}_{ii}+{r}_{ij}\le 1,i=1,2,i\ne j.$$

(2)

Managers’ limited attentional and cognitive capabilities may limit the ability of a firm to search for and learn from knowledge sources. We set the number of units of managerial time and effort available to the firm equal to one, which is the minimum strictly necessary to ensure that the firm has sufficient resources to fully use one innovation mechanism, either internal or external knowledge sourcing. A firm is likely to endure production disadvantages when conducting R&D to produce innovation and imitation. These disadvantages could result from the joint use of managerial inputs. In that case, its R&D production process involves diseconomies of scope. The cognitive and informational constraints imposed on firm management are exacerbated as firms increase the scope of their innovation activities. Scope diseconomies can offer each firm a difficult choice. In the extreme, each firm may have to choose between different innovation mechanisms representative of activities firms use to access knowledge in a discriminating fashion.

Next, we briefly describe the first-stage solution of the model. The solution strategy for determining δ_i, δ_j, r_ii, r_jj, r_ij, and r_ji will be useful to eliminate variables to obtain X_i as a function of variables δ_i, δ_j (and x_i, x_j) alone.

First-Stage Solution

We concentrate on the first stage of the game. Firms choose δ_i, δ_j, r_ij, and r_ji first. They make these choices by relying on the fact that both firms are rational and strategic. Surely, rational behavior implies that any resource allocation choice (r_ii, r_ij) = (1 − r_ij, r_ij), i = 1, 2, i ≠ j, from (2). Reducing managerial resource losses is essential to achieve the full profit potential. Firm i will make its choices taking into account its best guess as to firm’s best replies to the choices of δ_i and r_ij.

The equilibrium values of the model are found in a backward fashion. After deriving firms’ reduced-form profit functions π_i(δ_i, δ_j, r_ij, r_ji, x_i, x_j) and best-responses x_i to choices δ_i, δ_j, r_ij, and r_ji in stage 2, i = 1, 2, i ≠ j, firms choose those δ_i, δ_j, r_ij, and r_ji options which turn out to maximize their profits. The stage 1 equilibrium amount of firm ’s managerial attention allocated to external sources of knowledge is r_ij = δ_iδ_j. We can find firms’ optimal allocations of managerial resources given their optimal values of research approaches (and vice versa). Assume that δ_i is the optimal level of firm ’s research approach (and that δ_j and r_ji are optimal values from the perspective of firm j. We can prove that r_ij = δ_iδ_j in equilibrium by the method of contradiction. Suppose that r_ij ≠ δ_iδ_j. Clearly, choosing r_ij > δ_iδ_j is not optimal from firm ’s standpoint for a small decrease in r_ij such that ${r}_{ij}>{r}_{ij}^{\prime }>{\delta}_i{\delta}_j$ will increase the coefficient of x_i in X_i as r_ii = 1 − r_ij and firm i ’s individual profit. Any small change in r_ij would leave the coefficients of x_j and x_i in X_j unchanged. Choosing r_ij < δ_iδ_j will decrease the coefficient of x_j in X_i. But then, firm i could increase its profit by decreasing the coefficient of x_i in X_j without changing the coefficients in X_i through a small decrease in δ_i. So, the initial value of δ_i would not be part of the optimal solution to firm ’s profit maximization problem. The assumption that r_ij = δ_iδ_j is false must be wrong, so r_ij = δ_iδ_j must be true. There is a directly proportional relationship between r_ij and δ_iδ_j in equilibrium.

The direct gains (i.e., knowledge assimilation or cost reductions) from accessing outside sources of R&D are not costless. Firms may be adversely affected when conducting in-house R&D activities or bear costs in terms of loss of in-house R&D performance whenever they select broader (more similar) research approaches to keep them closer connected with their R&D environment. To put it another way, the efficiency coefficient of internal R&D output Min{1, r_ii} is 1 − δ_iδ_j < 1 only if δ_i > 0. Instead, 100% productivity of internal R&D is possible if firm i adopts the firm-specific R&D orientation δ_i = 0, which is then completely implemented with the support of managerial resources. Firm i has to take all available managerial resources and achieve maximum innovative results with them. It is worth noting that models of absorptive capacity in terms of research approaches traditionally assume that the choice of research design is purely strategic, so there are no-cost reasons for choosing one line of research to pursue rather than another (Katsoulacos & Ulph, 1998; Kamien & Zang, 2000; Wiethaus, 2005). On the directions for further research, Kamien and Zang (2000) suggest that it would be more realistic to suppose that R&D costs depend on the R&D approach chosen though.

Thus, X_i = (1 − r_ij)x_i + r_ijβx_j from expressions (1) and (2) alongside the stage 1 equilibrium considerations above or substituting δ_iδ_j in this equation for the effective level of cost reduction, we get

$${X}_i=\left(1-{\delta}_i{\delta}_j\right){x}_i+\beta {\delta}_i{\delta}_j{x}_j,i=1,2,i\ne j.$$

(3)

Firm ’s non-cooperative optimal choice of R&D approach for a given δ_j in stage 1 can be determined as solution to the problem of maximizing π_i(δ_i, δ_j, x_i, x_j) with respect to δ_i. The first-order conditions for each firm can be combined to form a system of two equations and two unknowns, the optimal research approach levels δ_i and δ_j. The profit maximization problem of firm i specified by defining each firm’s effective R&D level as in (3) is equivalent to the original problem of firm i specified in terms of representation (1) and resource constraint (2). Therefore, optimizing the individual profit of firm i only over δ_i is equivalent to the original problem where firm i optimizes over δ_i and r_ij simultaneously. The relation between the optimal solution of the original problem and the solution of its equivalent problem is clear as r_ij = δ_iδ_j.

The first stage of the game can be subsumed under a new first stage in which firms choose R&D approaches δ_i and δ_j. Any equilibrium level of managerial resource to look for new ideas outside a firm, r_ij, can be recovered by setting r_ij = δ_iδ_j. It thus appears appropriate to henceforth posit the representation of firm ’s effective R&D in (3). We propose a truncated three-stage game with the same timeline and strategic choice variables as in Kamien and Zang (2000) and Wiethaus (2005). In the game’s first stage, the two firms choose simultaneously their research approaches, and in the second stage, their levels of R&D output. Firms sell homogenous products to the final consumer in the third stage of the game, and the output level of each duopolist is chosen non-cooperatively. Kamien and Zang’s, Wiethaus’, and our duopoly model set-ups differ in the firm’s effective level of cost reduction.

In representation (3), there is a dilemma of searching for new knowledge internally or externally that each firm i faces in its choice of R&D approach δ_i. For absorption of external R&D, one needs the generality of firms’ R&D approaches, while for exploitation of internal innovative possibilities, the specificity of firms’ R&D approaches is more important. The economic dilemmas facing firms’ managers when there is a choice among multiple R&D approaches as illustrated in (3) can be grounded on one motive for substitutability between different mechanisms to innovation, that is, tight cognitive resource constraints and diseconomies of scope associated with using different innovation mechanisms in combination.

To account for representation (3) and the underlying substitutability between internal and external R&D, we focus on the case where there are diseconomies of scope which are managerial in nature and possibly large ones. The simultaneous use by a firm of different mechanisms to access knowledge implies a decrease of its innovative output and effective cost reduction as follows. If either δ_i = 0 or δ_j = 0, firm i ’s effective R&D activity turns into own R&D: X_i = x_i. The choice of δ_i = 0 reflects the preference of some firms for closed innovations as part of their innovation strategy. A “closed” innovation is an innovation developed by the firm itself without incorporating external knowledge. That innovative firm follows a strategy of relying exclusively on internal R&D resources and protecting systematically its own intellectual property from its rivals. When compared with the case of joint production of innovation and absorption, a firm specializing in innovation generates more unit-cost reduction from given x_i and so more R&D output level x_i and effective R&D level X_i, with the same fixed amount of production inputs than would a firm producing innovation and imitation. It becomes clear that joint production entails a loss, so diseconomies of scope in R&D activities are present in (3).

Consider a situation where firm i chooses the narrow firm-specific R&D orientation δ_i = 0. When compared with a broad orientation, a narrower one provides the advantage of higher efficiency of internal R&D, in the sense that the firm’s technological improvement obtained with any level of its own R&D expenditures is higher. However, a narrower orientation decreases the flexibility of the firm as to the accessibility of external knowledge. The disadvantage of the purely idiosyncratic orientation is that the level of effective cost reduction which is attained by the firm coincides uniquely with its own R&D output. Conversely, firms’ R&D orientations that are broad, i.e., δ_i > 0 and δ_j > 0, imply a favorable opportunity for capturing the technological opportunities arising from external sources of innovation. There is connectedness between firms if and only if both firms select broad orientations.

Results

In this section, we analyze how business conduct in a competitive setting may affect the choice of research approach under the assumption of scope diseconomies, and we explore how optimal R&D approaches and managerial resource allocation choices determine R&D appropriability and are affected by changes in exogenous spillovers. We establish the comparative-statics implications of our model.

We need a few considerations and definitions before stating the comparative-statics results of this section. Firms have incentives to manage the flows of information to and from competitors. The ability to protect proprietary technological knowledge from leaking to other firms depends on the firm’s innovation activities such as the choice of research approach, and the appropriability conditions in the industry. Legal protection is assumed to be an industry variable, rather than a firm-specific characteristic. Through their research approaches, firms are able to affect their appropriation capabilities. We only use strategic protection through the choice of (a narrower) research approach as a firm-level variable. The strategic choice variable R&D approach denotes generality of a firm’s R&D agenda. The larger (smaller) the δ_i, the more general (specific) is the R&D approach pursued by firm i. Some models of industrial organization suggest that firms attempt to manage the external information flows, trying to maximize incoming spillovers while at the same time minimizing outgoing spillovers (Cassiman, Perez-Castrillo, & Veugelers, 2002; Martin, 2002; Amir, Evstigneev, & Wooders 2003).

Spillovers are commonly assumed to be a parameter β that is exogenously given. One can interpret β as a parameter that is inversely related to appropriability or the degree of patent protection. Patenting constitutes an appropriation mechanism of R&D results, so if patent protection becomes stronger (weaker), then β decreases (increases). Let institutional appropriability in the industry be defined by the difference 1 − β. The portion 1 − β of new technology is assumed to be protected by patents and thus can effectively be controlled by its legal owner. From (3), total spillover per unit of R&D output generated is defined by the product βδ_iδ_j. Hence, we define total technological knowledge protection as 1 − βδ_iδ_j. The strength of firm ’s appropriability strategy is implicitly defined by protection 1 − βδ_iδ_j. By the representation of firm ’s effective R&D in (3), a decrease in the degree of generality of firm i ’s research approach, δ_i, by reducing firm i ’s knowledge leakage to firm j, increases the appropriability of the returns from the innovation by the innovator, firm i.

The stage 1 market equilibrium properties of the model are reported next.

Proposition 1

Consider the non-cooperative equilibrium with interior solution for R&D approaches, 0.57735 ≤ δ ≤ 1. We have the following:

δ(β) is everywhere decreasing in β.

β (δ(β))² is everywhere increasing in β.

Indeed, there is an increasing relationship between δ and (1 − β); there is an increasing relationship between δ and (1 − βδ²).

Discussion

We now explain the meaning of the findings of this study. Proposition 1 states that the equilibrium value of the research approach δ is monotonically decreasing in the exogenous spillover parameter β. The research approach δ decreases in β from 1 down to the intermediate level δ = 0.57735 at β = 1. This result implies that the generality of R&D performed by the firm is an increasing function of legal appropriability. Furthermore, the total spillover βδ² is increasing in β despite the decreasing behavior of δ. These two results imply that there is a positive association between δ and (1 − βδ²). Hence, the degree of R&D generality of firms’ research approach is positively related to the level of appropriability in the industry for the entire range of the spillover parameter.

Firms substitute internal knowledge sources for external sources as exogenous spillovers increase when they face constraints on managerial attention that are exacerbated by increased firm scope. These firms increase the scope of their innovation activities and endure R&D production disadvantages by developing and maintaining internal and external links to knowledge sources concurrently. The fear of knowledge spillovers leads firms to rely more on internal R&D even if the innovation mechanism of sourcing new technology internally is less productive than external technology sourcing activities; i.e., the efficiency coefficients of x_i and x_j in representation of X_i in (3) are such that 1 − δ² < δ² for δ > 0.70711, and the choices of research designs condition the dilemmas of innovation versus absorption that firms face. The higher the exogenous spillover rate, the more firms will need to control their knowledge leakages; i.e., connectedness between firms δ² is decreasing in β. As a result, the coefficient of x_i in X_i, 1 − δ², is increasing in β and the coefficient of x_j, βδ², is also increasing despite the decreasing behavior of δ. Non-cooperative firms manage to implement their equilibrium research approaches using managerial attention with the result that total knowledge protection is high enough. A positive R&D generality–total appropriability relationship can only occur if the overall level of appropriability is sufficiently high. Of course, if δ is decreasing in β, institutional appropriability always increases R&D generality.

To analyze firms’ decisions leading to the main results (Proposition 1) of this paper, we consider the underlying effects of R&D design on a firm’s profits. The decomposition of research approach incentives implies

$$\frac{\partial {\pi}_i}{\partial {\delta}_i}=\frac{\partial {q}_i}{\partial {\delta}_i}+\frac{\partial {q}_i}{\partial {x}_j}\frac{d{x}_j}{d{\delta}_i},i=1,2,i\ne j.$$

(4)

The direct effect of research design on profits, $\frac{\partial {q}_i}{\partial {\delta}_i}$, is negative provided that the amount of knowledge produced by a firm is higher than the amount of knowledge received from a rival, or β < 1 as x_i = x_j. The direct effect comes through marginal production cost increase; i.e., a broader approach increases the firm’s ability to identify, assimilate, and exploit new knowledge from an external source less than reduces the efficiency of internal R&D, making the firm to produce less final consumer goods. The less a firm produces at a higher marginal production cost, the less its profits. When the research approach level δ_i increases, the resulting increase in firm ’s connectedness to external sources of technological knowledge is offset by a simultaneous reduction in the efficiency of own R&D output. If β = 1, then $\frac{\partial {q}_i}{\partial {\delta}_i}=0$. An indirect effect on a firm’s profit is also at work, i.e., $\frac{\partial {q}_i}{\partial {x}_j}\frac{d{x}_j}{d{\delta}_i}$. This is the strategic effect: a broader research design reduces the profitability of a rival firm’s R&D investments by lowering appropriability and efficiency of internal R&D as $\frac{d{x}_j}{d{\delta}_i}<0$ for 0 < δ_j ≤ 1, having a positive effect on the firm’s profit provided that the appropriability term $\frac{\partial {q}_i}{\partial {x}_j}$ is negative. When the research design level δ_i increases, the resulting reductions in a rival’s R&D output due to lower appropriability and internal R&D efficiency reinforce each other. Only if δ_j = 0, $\frac{d{x}_j}{d{\delta}_i}=0$. The symmetric partial total derivative $\frac{\partial {\pi}_i}{\partial {\delta}_i}$ and each of its component derivative expressions can be found in the Appendix. The derivative $\frac{\partial {\pi}_i}{\partial {\delta}_i}$ in (4) shows a trade-off between these two effects of a change of δ_i on profits as long as the appropriability term is negative, or to put it better, non-positive.

The trade-off between connectedness and appropriability can only exist everywhere in the spillover parameter space if the appropriability term is non-positive, meaning that the overall level of appropriability is sufficiently high for every β:

$$\frac{\partial {q}_i}{\partial {x}_j}=\frac{1}{3}\left(-1+\left(1+2\beta \right){\delta}_i{\delta}_j\right)\le 0,$$

(5)

or equivalently, ${\delta}_i{\delta}_j\le \frac{1}{1+2\beta\ }$, which implies $1-\beta {\delta}_i{\delta}_j\ge 1-\frac{\beta }{1+2\beta\ }$. This requirement in (5) is satisfied by Proposition 1. Supposing that the appropriability term is negative, firms have to trade off the negative direct effect against the positive strategic effect. By Proposition 1, the positive strategic effect is more likely to dominate the direct effect at a given research approach level $0.57735<{\delta}_i^{\prime }<1$ for low levels of β. The opposite relative intensity of the two effects at the same level of δ_i is more likely to hold for high levels of β. Firms’ R&D design choices depend on the relative intensity of the direct and strategic effect. Firms respond by choosing a more similar R&D approach than ${\delta}_i^{\prime }$ when the strategic effect dominates the direct effect. If the opposite holds, firms choose a more firm-specific approach than ${\delta}_i^{\prime }$. This means that firms adopt narrower research approaches when the spillover level β is high and broader approaches when β is low; in other words, variables δ_i and β move in opposite directions in a monotonic indirect relationship in the spillover parameter space.

Increasing R&D generality is the approach firms use to establish external linkages and build connectedness to external sources of knowledge. R&D generality can help firms to take knowledge from external sources. However, it also leads to unintended knowledge spillovers, limiting firms’ ability to protect their own knowledge. We find a tension between the revelation of some parts of proprietary knowledge by adopting a more general R&D approach and the protection of one’s own knowledge from being copied by competitors by adopting a more specific approach. This tension comes to live as a paradox that firms understand. With higher exogenous spillovers, the fear of providing costless benefits to rivals leads to a decrease in the level of generality of firms’ research approach in order to reduce technology transfer to rivals.

Our findings reflect some concerns of firms that are likely to become “open” innovators by engaging with external sources of innovation but face the “paradox” of openness. Managers make their firms open to different external organizations in order to access external knowledge in their innovation activities, but also have to protect their own knowledge against being imitated by rivals. The more open these innovative firms are in terms of their external knowledge exchange, the more they see the protection of their core knowledge as relevant. There is some evidence supporting the claim that there is a positive association between openness for innovation and appropriability of innovation benefits, and we interpret our theoretical result on the relationship between the degree of R&D generality and the level of appropriability to be consistent with this empirical result.

The measures of generality of the R&D approach and openness of firms’ external search strategies can both be based on external sources of innovation. The generality of the R&D approach is empirically proxied by Kaiser (2002). In his econometric analyses of German firms, it is assumed that the more general a firm’s research approach is, the more heterogeneous its information sources are. The proxy variable is constructed as the number of information sources a firm regards as important for the innovation process. The available information sources include customers, suppliers, competitors, and several other public and private institutions. In turn, Laursen and Salter (2006) develop two concepts indicating the openness of individual firms’ external search strategies. The first concept refers to external search breadth, and it is defined in terms of the number of external sources that firms rely on their innovative activities. The second concept refers to external search depth, which is defined as the extent to which firms draw deeply from the different external sources. The authors use two proxies as measures of openness in terms of external search strategies, breadth and depth, that reflect the importance of external sources to innovative performance too. The list of external sources of innovation includes suppliers, clients, competitors, and general institutions in the innovation system. The findings of this study imply that the firms using external sources of knowledge in a larger number and to a larger extent tend to be more open than others.

Conclusion

In this paper, we have explored how substitutability between internal and external R&D affects the market behavior of innovating firms in terms of research approaches and managerial resource allocations under competition. We sought to elucidate the issue about the relationship between appropriability and openness. A body of evidence supports the view that searching for knowledge internally and externally is a substitutive R&D activity. To examine the theoretical implications of this view, we set up a duopoly model which incorporates the assumption of substitutability arising between internal and external R&D activities due to managerial diseconomies of scope in firms investing in various innovation activities, thereby extending the existing work on strategic R&D investments with exogenous spillovers and endogenous absorptive capacity. We are aware that there may only be a few game-theoretic works addressing this topic. By taking into account in our model the assumption of internal and external sources of knowledge as substitutes, the qualitative results of earlier literature on R&D with exogenous spillovers or endogenous absorptive capacity relying on research designs no longer survive. Empirical work has been developed that validates the predictions of our theory.

The main results of this paper can be summarized as follows. Our model predicts that when the joint pursuit of internal and external knowledge search activities produces diseconomies of scope, firms will respond by relying more on internal R&D and less on external R&D as exogenous spillovers increase. We concentrate our attention on the R&D generality of firms’ research approaches as a way to access to external knowledge sources. We find a tension between the knowledge revelations resulted from the generality of R&D performed by a firm and the protections of proprietary knowledge facilitated by the specificity of R&D performed by it that emerges as a paradox that firms confront. The finding of a positive relationship between the choice of a firm to be connected with external sources of knowledge and the appropriability of its innovation benefits is regarded as being consistent with empirical results of a paradox of openness that firms open to external knowledge sources understand.

We consider the implications of the hypothesis of substitutability between internal and external R&D from the viewpoint of public policy toward R&D in a second-best situation. The economy operates under second-best conditions because of imperfect competition in the product market. It is well-known that knowledge spillovers decrease the level of R&D per firm because of a free-rider or disincentive effect. The social returns from R&D output are greater than the private benefits captured by innovating firms. The existence of this market failure that leads the private sector to underinvest in R&D is one of the main rationales for government R&D subsidies. To the extent that there exist R&D appropriability problems, conventional wisdom stresses the incentive and ability of RJVs to internalize R&D externalities in the market, at least, for high spillovers. Strengthening IPRs is another way to restore private R&D incentives and close the gap between social and private returns to R&D.

Contrary to prior literature, we can find that when there are scope diseconomies in knowledge searching, oligopolistic rivalry in innovation will sometimes support socially second-best R&D. In this welfare analysis, we use the second-best welfare function as the welfare criterion. It can be shown that the stage 1 equilibrium of the game is either an interior solution or a corner solution. As it happens, the equilibrium level of R&D output per firm under independent R&D induces the second-best R&D for society if firms adopt the efficient, narrow firm-specific R&D approach. The corner equilibrium entails a higher level of individual R&D output relative to the interior equilibrium. Welfare may rise if a sub-optimally low level of R&D output is increased by increasing the efficiency of R&D efforts.

The competitive innovation game can ensure the second-best equilibrium outcome irrespective of whether exogenous R&D spillovers are large or small. As such, there is no justification for government intervention in the market to achieve the second-best social optimum. Under such conditions, no form of market failure exists associated with R&D and innovation, so government subsidies to R&D play no role in fostering technological progress. If public funds are costless, then government R&D subsidization is a first-best policy. First-best policies do not need to be applied to improve welfare in the second-best world. The best outcome that any such policy can achieve is the second-best outcome. We consider two other policy approaches that are aimed at correcting R&D market failures through market forces and government intervention in the form of patents granted to industry, to conclude that the former is at best a second-best solution to imperfections in the market. The rank order of the latter potential policy is higher if any. Reliance on market power and research joint ventures (RJVs) to provide incentives to innovate and intellectual property rights (IPRs) cannot be an effective means of correcting market failures. The corner equilibrium outcome undermines the welfare case for promoting RJVs and defeats the case for strengthening IPRs.

Some important limitations of this study are worth mentioning. First, specification of effective cost reduction is based on a particular amount of managerial attention allocated to search activities. This constrained managerial resource endowment may have idiosyncratic characteristics; therefore, the suggested specification of effective cost reduction and its theoretical implications must be validated against other cases where the resource constraint is either relaxed or made more binding. Specifically, the theoretical focus on one single managerial environment limits generalizability. Second, the fully non-cooperative case of R&D game that we considered and its theoretical outcomes must be validated against other cases with different modes of organizing R&D such as R&D cooperation, where firms cooperate in the choice of their R&D approaches and levels. One may not find in R&D cooperation the tension between OI and appropriability as characterized by a positive relationship between R&D generality and appropriability level. The selected competitive mode of R&D organization thus limits generalizability outside the organizational context that was investigated. Therefore, future research could extend the findings of this study with cooperative-behavior elements and investigate collaborative forms of business conduct that competing firms might adopt when undertaking R&D and sharing R&D results, for instance, following Kamien and Zang (2000). Such studies would give insight into the actions that managers take in response to tensions between OI and appropriability in collaborative arrangements.

Appendix

This appendix contains a detailed description of the model apart from the representation of the effective R&D level, which is given in the text of the paper, and the proof of Proposition 1, which is accompanied by figures containing graphs of functions.

A. Model

Basic Model

Consider an industry with two firms indexed by i = 1, 2 producing a homogeneous good. As for the cost terms, firms can reduce their constant unit cost of production A by the amount of their effective R&D level X_i, but fixed costs depend on own R&D and also on a scarce managerial resource. The i th firm’s effective R&D activity is given by (3).

Firms are restricted in size to employ no more than a fixed amount of managerial resources. The amount of the managerial resource r available to each firm is assumed to equal unity, and its price is denoted by p_r, so this component of fixed costs is p_r. The R&D technology they employ exhibits diminishing returns, as each firm’s cost of R&D investment x_i is quadratic, $\left(\gamma /2\right){x}_i^2$.

Each firm seeks to maximize individual profit

$${\pi}_i=\left(a-{q}_i-{q}_j\right){q}_i-\left(A-{X}_i\right){q}_i-\frac{\gamma }{2}{x}_i^2-{p}_r\kern1.25em i=1,2,i\ne j,$$

(6)

where a is the highest consumer reservation price, and q_i and q_j quantities produced by firm i and firm j, respectively. The demand function P(Q) = a − q_i − q_j determines the market price P as a linear function of the market quantity Q = q_i + q_j.

Third-Stage Equilibrium Results

The method of backward induction is adopted to find the equilibrium values of the model, and therefore, the quantity choices at the final stage of the game are derived first. From the profit function (6), we obtain optimal output quantities as

$${q}_i=\frac{1}{3}\left(a-A+2{X}_i-{X}_j\right),i=1,2,i\ne j.$$

(7)

Equilibrium Results in R&D Competition

Characterization of the Optimal Level R&D Investment

Here, we concentrate on the competitive solution at both the initial stage of selection of R&D approaches and the second stage of choice of R&D investments to deduce the symmetric-stage Nash equilibria. We consider an industry with two ex ante identical firms, so it is natural to assume that firms adopt the same actions in equilibrium. Given the equilibrium level of output (7) at the third stage, the i th firm’s profit function of the non-cooperative game’s second stage can be expressed as

$${\pi}_i={q}_i^2-\frac{\gamma }{2}{x}_i^2-{p}_r,i=1,2.$$

(8)

The first-order condition for a maximum of π_i with respect to x_i, i = 1, 2, for an interior solution can be written as

$$\frac{\partial {\pi}_i}{\partial {x}_i}=2{q}_i\frac{\partial {q}_i}{\partial {x}_i}-\gamma {x}_i=0,i=1,2.$$

(9)

Note that, from (3) and (7),

$$\frac{\partial {q}_i}{\partial {x}_i}=\frac{1}{3}\left(2-\left(2+\beta \right){\delta}_i{\delta}_j\right).$$

(10)

For an interior solution, we must have ∂q_i/∂x_i > 0. From (9), we deduce the equilibrium level of investment

$${x}_i=\frac{2\left(a-A\right)\left(2-\left(2+\beta \right){\delta}_i{\delta}_j\right)}{\varPsi_i},i=1,2,i\ne j,$$

(11)

where

$${\varPsi}_i=9\gamma -2\left(2-\left(2+\beta \right){\delta}_i{\delta}_j\right)\left(1+\left(-1+\beta \right){\delta}_i{\delta}_j\right).$$

(12)

The second-order condition with respect to x_i is satisfied for γ > 8/9, and to see this, just consider the most stringent situation, at β = 0 if δ = 1. By γ > 8/9, the denominator of the symmetric equilibrium R&D investment in (11) is non-negative. The stability conditions hold at the non-cooperative R&D equilibrium for γ > 4/3.

Characterization of the Optimal Research Approach

The first-order condition with respect to the R&D approach δ_i for an interior solution, ∂π_i/∂δ_i = 0, can be described by a direct effect and a strategic effect as follows:

$$\frac{\partial {\pi}_i}{\partial {\delta}_i}=2{q}_i\left(\frac{\partial {q}_i}{\partial {\delta}_i}+\frac{\partial {q}_i}{\partial {x}_j}\frac{d{x}_j}{d{\delta}_i}\right),i=1,2,i\ne j.$$

(13)

Note that the strategic effects ∂π_i/∂x_i, i = 1, 2, are zero by the second-stage first-order conditions (9) and so are omitted here. To evaluate the overall effect of a change of δ_i on profit, all the derivatives on the right side of (13) must then be derived. To begin with, the direct effect,

$$\frac{\partial {q}_i}{\partial {\delta}_i}=\frac{1}{3}\left(\left(-\left(2{x}_i-{x}_j\right)+\beta \left(2{x}_j-{x}_i\right)\right){\delta}_j\right),i=1,2,i\ne j,$$

(14)

is non-positive in δ_i if

$$\left(2+\beta \right){x}_i\ge \left(1+2\beta \right){x}_j,i=1,2,i\ne j.$$

(15)

Next, for the symmetric case, we have the familiar appropriability term

$$\frac{\partial {q}_i}{\partial {x}_j}=\frac{1}{3}\left(-1+\left(1+2\beta \right){\delta}^2\right).$$

(16)

Finally, the effects of a change in δ_i on the rival’s optimal R&D investment in the game’s second stage, x_j, are referred by the derivative

$$\frac{d{x}_j}{d{\delta}_i}=-\frac{2\left(a-A\right)\left(2+\beta \right){\delta}_j\left(9\gamma -\right)}{\varPsi_i^2},i=1,2,i\ne j.$$

(17)

The sign of dx_j/dδ_i is determined by the sign of the numerator of the evaluated derivative since the denominator is positive. Hence, dx_j/dδ_i is non-positive.

Let π_i(δ_i, δ_j) denote the first-stage profit function of firm i. Suppose that the symmetric first-order condition ∂π_i(δ, δ)/∂δ_i = 0 is satisfied at δ = δ^′ > 0. Profit π_i is maximized at δ^′ since ∂π_i(δ, δ)/∂δ_i is positive between 0 and δ^′ and goes from positive to negative at δ^′. Hence, there is an interior equilibrium level δ^′ > 0.

There exists another symmetric Nash equilibrium at δ_i = δ_j = δ = 0. The first and second derivatives of π_i(δ_i, δ_j) with respect to δ_i are all zero at δ = 0, and both derivatives equal zero on both sides of that point.

B. Proof

Proof of Proposition 1

Consider first the behavior of the interior solution for R&D approach in relation to the spillover parameter β under R&D competition. Then, consider the behavior of the function total spillover βδ² as the value of the spillover parameter increases.

Suppose that δ_j ≠ 0 (as there is a subgame perfect Nash equilibrium at the extreme δ = 0), δ_i = δ_j = δ, and γ > 8/9 (by the SOC with respect to x_i). We substitute for ∂q_i/∂δ_i from (14), x_i from (11), ∂q_i/∂x_j from (16), and dx_j/dδ_i from (17) into (13) to get, after collecting terms,

$$\mathit{\operatorname{sgn}}\ \partial {\pi}_i/\partial {\delta}_i=\mathit{\operatorname{sgn}}\ \left(-1+\beta \right)\left(2-\left(2+\beta \right){\delta}^2\right)\varPsi -\left(-1+\left(1+2\beta \right){\delta}^2\right)\left(9\gamma \left(2+\beta \right)-2{\left(2-\left(2+\beta \right){\delta}^2\right)}^2\left(-1+\beta \right)\right)$$

(18)

Note here that the sign of the expression ∂π_i/∂δ_i is determined by the sign of the numerator of the evaluated derivative as the denominator is always positive.

Let $\overset{\sim }{X}={\delta}^2$. Setting ∂π_i/∂δ_i = 0 produces a cubic equation of the form:

$$\overset{\sim }{a}{\overset{\sim }{X}}^3+\overset{\sim }{b}{\overset{\sim }{X}}^2+\overset{\sim }{c}\overset{\sim }{X}+\overset{\sim }{d}=0,$$

(19)

where

$$\overset{\sim }{a}=-2\left(-1+\beta \right){\left(2+\beta \right)}^2+2\left(-1+\beta \right)\beta {\left(2+\beta \right)}^2+2{\left(2+\beta \right)}^2\left(1+2\beta \right)-2\beta {\left(2+\beta \right)}^2\left(1+2\beta \right),$$

(20)

$$\overset{\sim }{b}=8\left(-1+\beta \right)\left(2+\beta \right)-8\left(-1+\beta \right)\beta \left(2+\beta \right)-4{\left(2+\beta \right)}^2+4\beta {\left(2+\beta \right)}^2-8\left(2+\beta \right)\left(1+2\beta \right)+8\beta \left(2+\beta \right)\left(1+2\beta \right),$$

(21)

$$\overset{\sim }{c}=-8\left(-1+\beta \right)+8\left(-1+\beta \right)\beta +16\left(2+\beta \right)-16\beta \left(2+\beta \right)+8\left(1+2\beta \right)-8\beta \left(1+2\beta \right)-9\left(2+\beta \right)\gamma +9\beta \left(2+\beta \right)\gamma +9\left(2+\beta \right)\left(1+2\beta \right)\gamma,$$

(22)

$$\overset{\sim }{d}=-16+16\beta +18\gamma -18\beta\ \gamma -9\left(2+\beta \right)\gamma .$$

(23)

To derive the algebraic solution of the cubic equation, we begin by calculating the discriminant of the polynomial equation and then reducing the polynomial to a depressed cubic equation. Derivation of the roots involves the application of Vieta’s substitution to the depressed cubic.

$$\Delta =18\overset{\sim }{a}\ \overset{\sim }{b}\ \overset{\sim }{c}\ \overset{\sim }{d}-4{\tilde{b}}^3\ \overset{\sim }{d}+{\tilde{b}}^2\ {\tilde{c}}^2-4\overset{\sim }{a}\ {\tilde{c}}^3-27\ {\tilde{a}}^2\ {\tilde{d}}^2$$

(24)

$${\Delta }_0={\tilde{b}}^2-3\overset{\sim }{a}\ \overset{\sim }{c}$$

(25)

$$q=\left(2{\tilde{b}}^3-9\overset{\sim }{a}\ \overset{\sim }{b}\ \overset{\sim }{c}+27{\tilde{a}}^2\ \overset{\sim }{d}\right)/\left(27{\tilde{a}}^3\right)$$

(26)

$$p=\left(3\overset{\sim }{a}\ \overset{\sim }{c}-{\tilde{b}}^2\right)/\left(3{\tilde{a}}^2\right)$$

(27)

$$W=-\frac{q}{2}+\sqrt{-\frac{\Delta }{108{\tilde{a}}^4}}=-\frac{q}{2}+\sqrt{{\left(\frac{q}{2}\right)}^2+{\left(\frac{p}{3}\right)}^3}$$

(28)

$$Z={W}^{1/3}$$

(29)

$$Y=Z-\frac{p}{3Z}$$

(30)

$$\overset{\sim }{X}=Y-\frac{\tilde{b}}{3\overset{\sim }{a}}$$

(31)

Hence, the symmetric solution

$$\delta =\sqrt{\overset{\sim }{X}}.$$

(32)

To determine the number of real and complex roots, we look at the discriminant. For 0 < β < 1, ∆ < 0, so there exists one real cube root (Eq. (31)) and 2 complex roots ⇒ the interior solution δ (Eq. (32)). For β = 0 ∨ 1, ∆ = 0 ∧ ∆₀ = 0, so there is a triple root ⇒ δ = 1 ∨ 0.57735, respectively.

The interior symmetric solution δ as a function of 0 ≤ β ≤ 1, 0.57735 ≤ δ_i ≤ 1, is depicted next in Fig. 1 for (i) γ = 8/9, (ii) γ = 10/9, (iii) γ = 4/3, (iv) γ = 10, (v) γ = 100, (vi) γ = 1000, and (vii) γ = 10000 (using Mathematica 2.0 software).

Fig. 1

Interior symmetric R&D approach δ as a function of β

Likewise, Fig. 2 displays total spillover βδ². This figure shows that δ² always decreases with increasing rates of exogenous spillover but not enough to cancel it completely.

Fig. 2

Total spillover βδ² as a function of β

These results imply that the relationships between δ and (1 − β) and between δ and (1 − βδ²) are everywhere positive. To see this, one just has to define the derivatives $\frac{d\left(1-\beta \right),}{d\delta}$ and $\frac{d\left(1-{\beta \delta}^2\right)}{d\delta}$ as the ratios of two first derivatives of parametric equations, respectively $\frac{d\left(1-\beta \right)/ d\beta\ }{d\delta / d\beta}$ and $\frac{d\left(1-{\beta \delta}^2\right)/ d\beta}{d\delta / d\beta}$, and determine their signs (both positive).